On Stieltjes integrals and Parseval's equality for multiple trigonometric series

T. P. Lukashenko

Izvestiya. Mathematics , Tome 69 (2005) no. 5, pp. 1005-1024

Résumé

In this paper, it is proved that if a function $f$ from $\mathbb R^n$ to $\mathbb C$ is $2\pi$-periodic with respect to each variable and Lebesgue integrable on $T^n=[0,2\pi]^n$, a complex-valued additive segment function $\mathcal G$ is defined on all segments in $\mathbb R^n$ and is $2\pi$-periodic with respect to each variable, the point function $G$ corresponding to $\mathcal G$ is Lebesgue integrable on $T^n$, and the function $f$ is integrable with respect to $\overline{\mathcal G}$ in the Riemann–Stieltjes sense on all shifts of $T^n$, then Parseval's equality holds with the series not necessarily convergent, but summable by Riemann's method. Some results are also obtained on Parseval's equality for Fourier–Lebesgue–Stieltjes multiple trigonometric series.